Quite the opposite, much like when skydiving, going really fast without any close reference point makes everything stand still. And in space, there wouldn’t even be (very loud) atmospheric drag to physically remind you about what speed you’re actually going.
1) it's better for the plot and drama to have travel time. FTL in fiction is always analagous to some known terrestrial form of travel (usually ships and boats) and the limitations and parameters of FTL in a fictional universe shape the narrative in necessary ways.
2) it's assumed within the framework of the fictional universe that time dilation isn't taking place because the actual travel is occurring within an external frame of reference like "hyperspace" or a "warp field."
I believe the OP was referring to relativity - the closer to the speed of light you get the slower time appears to tick. So if you could travel at light speed you'd arrive at your destination immediately from your reference frame, but much slower from another person's.